Spectral Asymptotics for Non-self-adjoint Semiclassical Operators

University Of California-Los Angeles, Los Angeles CA

Investigators

Abstract

PI: James V. Ralston (for Hatrik), UCLA DMS-0304970 ABSTRACT This proposal presents problems in spectral theory of non-self-adjoint differential operators in the semi-classical regime. The proposer seeks precise estimates on the asymptotic behavior of the eigenvalues of small perturbations of self-adjoint operators on compact domains in several settings. In closely related projects he plans to apply recently developed techniques to the problem of finding asymptotics (counting functions) for the scattering poles associated with convex obstacles and the barrier top resonances for Schroedinger operators. This work studies the propagation of waves in settings where some form of dissipation or the possibility of propagation to infinity gives rise to waves which decay to zero as time increases. The rates of decay and frequency of these decaying modes are encoded in sequences of complex eigenvalues or resonances associated with these problems. Studying the behavior of these sequences can lead to better understanding of the relation between rates of decay and the underlying structure of the system. Such information has potential application in determining the interior structure of objects from their resonant frequencies in nondestructive testing.